The club skipper has taken full responsibility for kicking Neymar, although the midfielder and his manager Neil Lennon insist the decision to send him off was harsh.
The 28-year-old Brown was sent off for kicking Neymar in the 58th minute, a decision that Brown, along with manager Neil Lennon, deemed as harsh.
Following his dismissal, Celtic was unable to hold on for a valuable point in its bid to stun Barcelona for a second straight season, with Cesc Fabregas taking advantage of the Catalan side's numerical advantage 17 minutes later.
"I realize that I made a mistake in getting sent off. It's something that happened in the heat of the moment but I shouldn't have acted in this way," Brown told the club's official website. "I still think it was a harsh decision. However, I recognize that I made a mistake and put myself and my teammates in a difficult position. I need to live with that - and there is no one more disappointed than me about losing the match."
Neymar's role in the midfielder's dismissal has faced criticism from Lennon and Scotland manger Gordon Strachan, while Brown has questioned the Brazil star's reaction to the challenge.
"There was not a huge amount of contact at all, to be honest, but I have to accept responsibility for my actions - and I do," Brown said.
Brown will now miss Ajax's visit to Celtic Park on Oct. 22, but if his actions are deemed excessive by UEFA, he could also be forced to sit out the return clash with the Eredivisie side and the home game against AC Milan.